This work concerns a detailed review of data analysis methods used for remotely sensed images of large areas of the Earth and of other solid astronomical objects. In detail, it focuses on the problem of inferring the materials that cover the surfaces captured by hyper-spectral images and estimating their abundances and spatial distributions within the region. The most successful and relevant hyper-spectral unmixing methods are reported as well as compared, as an addition to analysing the most recent methodologies. The most important public data-sets in this setting, which are vastly used in the testing and validation of the former, are also systematically explored. Finally, open problems are spotlighted and concrete recommendations for future research are provided.